Obviously, as \(N \rightarrow \infty \) , for each fixed \(k\) , we have:
\(a(N, k) := \left[\frac{N(N+c)}{N-k}\right]^{1/2} \rightarrow 1,\quad N \rightarrow \infty .\) 
Multiply the \(k\) th component \(\hat{\mathbb {C}}_k\)  of the vector \(\hat{\mathbb {C}}\)  by \(a(N, k)\) . Apply Slutsky's theorem, [1]}, and use (REF ). Then we get:
\(\left[a(N, 1)\hat{\mathbb {C}}_1, \ldots , a(N, k)\hat{\mathbb {C}}_K\right] \Rightarrow \mathcal {N}_{m^2K}(0, I_{m^2K}).\) 
Taking the square of the Euclidean norm on both sides of (REF ), similarly to the proof of Corollary REF , we get (REF ).
